TAMU CHEN 304 - L11 (15 pages)

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L11



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L11

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Pages:
15
School:
Texas A&M University
Course:
Chen 304 - Chem Engr Fluid Ops
Chem Engr Fluid Ops Documents

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CBE341 Lecture 11 10 6 Constitutive equations for a flowing fluid What have we derived thus far in the course Till now I focused on deriving conservation equations by using the principle of mass conservation and Newton s law of linear momentum balance Let us have a recap of the equations I first formulated the integral mass balance equation See equation 36 of my notes Then I deduced the differential form of this balance commonly referred to as the continuity equation See equation 43 for a general case For the special case of an incompressible fluid we obtain equation 46 Then I started with Newton s law for linear momentum balance I derived the integral form of this balance applicable to an arbitrary control volume See equation 56 of my notes Then I introduced the notion of a deviatoric stress tensor see equations 117 and 118 Finally I converted equation 56 to a differential form referred to as the Cauchy s equations of motion see equations 129 and 136 We will now discuss the additional information needed before we can use the continuity equation and the Cauchy s momentum balance equations to find detailed solutions for the velocity profiles in some flow problems The continuity equation and the Cauchy s momentum balance equations are reproduced below for convenience r rv 0 t 43 rv rvv rg P t t 87 129 Or equivalently r v rv v rg P t t 136 Many flow problems involve only modest pressure drops and the volume change is small even for gases Hence even gas flow problems can be treated as incompressible fluid flow problems with little loss of accuracy For incompressible fluids equation 43 simplifies to v 0 46 Aside If we consider examples such as flow of a gas through a very long pipe with correspondingly large pressure drops and appreciable volume changes then we retain equation 43 Even in these problems the effect of compressibility in the momentum balance equations namely 129 or 136 turns out to be weak provided we are not talking about flows involving high Mach numbers close to unity or larger In chemical engineering we rarely deal with such problems and so we can ignore the effect of compressibility in equations 129 and 136 In this course where we want to illustrate the use of balance equations to solve flow problems we will only consider essentially incompressible flows So we will discuss the constitutive equations for the stress tensor only for the case of incompressible fluids It was already mentioned that the viscous stress tensor is symmetric Note that the viscous stress depends on the velocity gradient For example consider the so called plane Couette flow 88 According to our notations the viscous traction exerted by the upper plate on the fluid sandwiched between the two plates is given by t n t where n is the unit normal pointing from the entity on which the traction is being exerted in this case the fluid towards



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